Supplement: Risk factors for mortality among hospitalized patients with COVID-19

1 Dependencies and settings

# R packages
library("corrr") # 0.4.2
library("cowplot") # 1.0.0
library("dplyr") # 1.0.0
library("ggplot2") # 3.3.1
library("oem") # 2.0.10
library("gridExtra") # 2.3
library("knitr") # 1.28
library("kableExtra") # 1.1.0
library("magrittr") # 1.5
library("mice") # 3.9.0
library("pROC") # 1.16.2
library("purrr") # 0.3.4
library("rcompanion") # 2.3.25
library("rms") # 6.0.1
library("splines") # base
library("tibble") # 3.0.1
library("tidyr") # 1.1.0

# Settings
set.seed(123)
theme_set(theme_bw())

# Namespace clash
select <- dplyr::select

2 Data

2.1 Load

train_data <- readRDS("train_data.rds")  %>%
  filter(age >= 18) %>%
  filter(as.Date("2020-06-05") - index_date > 14)

2.2 Clean

# Recoding
train_data <- train_data %>% 
  mutate(
    ## "died" in add_deaths() should be an integer
    died = as.integer(died),
    
    ## Convert unknown or other/unknown to missing
    race = ifelse(race == "Other/Unknown", NA, race),
    ethnicity = ifelse(ethnicity == "Unknown", NA, ethnicity),
    sex = ifelse(sex == "Unknown", NA, sex),
    
    ## Oxygen saturation should have plausible values
    spo2 = ifelse(spo2 == 0, NA_real_, spo2),
    spo2 = ifelse(spo2 > 100, 100, spo2)
  ) %>%
  
  ## Add option for comorbidities in create_comorbidities() to be character
  rename(diabunc = diab) %>%
  mutate_at(
    c("ami", "chf", "pvd", "cevd", "dementia", "copd", "rheumd", "pud",
      "mld", "diabunc", "diabwc", "hp", "rend", "canc", "msld", "metacanc",
      "aids", "hypc", "hypunc", "asthma"),
    function (x) ifelse(x == 1, "Yes", "No")
  ) 

# Create "derived" variables
train_data <- train_data %>%
  mutate(
    calendar_time = as.numeric(index_date - min(index_date)),
    index_month = as.integer(format(index_date, "%m")),
    death_month = as.integer(format(date_of_death,"%m")),
    os_days = pmax(0, date_of_death - index_date),
    
    ## Categorize continuous vitals + labs
    ### BMI
    bmi_cat = case_when(
      bmi < 18.5 ~ "Underweight",
      bmi >= 18.5 & bmi < 25 ~ "Normal",
      bmi >= 25 & bmi < 30 ~ "Overweight",
      bmi >= 30 ~ "Obese",
      TRUE ~ NA_character_
    ),
    
    ### Blood pressure
    bp_cat = case_when(
      dbp < 80 | sbp < 120 ~ "Normal",
      dbp < 80 & (sbp >= 120 & sbp < 130) ~ "Elevated",
      dbp >= 80 | sbp >= 130 ~ "High",
      TRUE ~ NA_character_
    )
  )

# Function to add race/ethnicity variable to dataset (done after separately 
# imputing missing race + ethnicity information)
add_race_ethnicity <- function(data){
 data %>% mutate(
   race_ethnicity = case_when(
    race == "Caucasian" & ethnicity != "Hispanic" ~ "Non-Hispanic white",
    race == "African American" & ethnicity != "Hispanic" ~ "Non-Hispanic black",
    race == "Asian" & ethnicity != "Hispanic" ~ "Asian",
    TRUE ~ "Hispanic"
  ),
  race_ethnicity = relevel(factor(race_ethnicity), ref = "Non-Hispanic white")
 )
}

2.3 Label

# Labels
## Categorical variables
demographic_cat_vars <- tribble(
  ~var, ~varlab,
  "sex", "Sex",
  "race", "Race",
  "ethnicity", "Ethnicity",
  "division", "Geographic division"
) %>% 
  mutate(group = "Demographics")

comorbidity_vars <- tribble(
  ~var, ~varlab,
  "ami", "Acute myocardial infarction",
  "chf", "Congestive heart failure",
  "pvd", "Peripheral vascular disease",
  "cevd", "Cerebrovascular disease",
  "dementia", "Dementia",
  "copd", "COPD",
  "rheumd", "Rheumatoid disease",
  "pud", "Peptic ulcer disease",
  "mld", "Mild liver disease",
  "diabunc", "Diabetes (no complications)",
  "diabwc", "Diabetes (complications)",
  "hp", "Hemiplegia or paraplegia",
  "rend", "Renal disease",
  "canc", "Cancer",
  "msld", "Moderate/severe liver disease",
  "metacanc", "Metastatic cancer",
  "aids", "AIDS/HIV",
  "hypunc", "Hypertension (no complications)",
  "hypc", "Hypertension (complications)",
  "asthma", "Asthma"
) %>%
  mutate(group = "Comorbidities")

vital_cat_vars <- tribble(
  ~var, ~varlab,
  "smoke", "Smoking",
  "bmi_cat", "Body Mass Index (BMI)",
) %>%
  mutate(group = "Vitals")

cat_vars <- bind_rows(demographic_cat_vars,
                      comorbidity_vars,
                      vital_cat_vars)

## Continuous variables
demographic_continuous_vars <- tribble(
  ~var, ~varlab,
  "age", "Age",
  "calendar_time", "Time since first case"
) %>%
  mutate(group = "Demographics")

vital_continuous_vars <- tribble(
  ~var, ~varlab,
  "bmi", "Body Mass Index (BMI)",
  "dbp", "Diastolic blood pressure",
  "sbp", "Systolic blood pressure",
  "hr", "Heart rate",
  "resp", "Respiration rate",
  "temp", "Temperature",
) %>%
  mutate(group = "Vitals")

lab_vars <- tribble(
  ~var, ~varlab,
  "alt", "Alanine aminotransferase (ALT)",
  "ast", "Aspartate aminotransferase (AST)",
  "crp", "C-reactive protein (CRP)",
  "creatinine", "Creatinine",
  "ferritin", "Ferritin",
  "d_dimer", "Fibrin D-Dimer",
  "ldh", "Lactate dehydrogenase (LDH)",
  "lymphocyte", "Lymphocyte count",
  "neutrophil", "Neutrophil count",
  "spo2", "Oxygen saturation",
  "pct", "Procalcitonin",
  "tni", "Troponin I",
  "plt", "Platelet count (PLT)",
  "wbc", "White blood cell count (WBC)"
) %>%
  mutate(group = "Labs")

continuous_vars <- bind_rows(
  demographic_continuous_vars,
  vital_continuous_vars,
  lab_vars
)

# Binary outcomes
binary_outcomes <- tribble(
  ~var, ~varlab,
  "died", "Died" 
)

# All variables
vars <- bind_rows(cat_vars,
                  continuous_vars)

get_var_labs <- function(v){
  vars$varlab[match(v, vars$var)]
}

3 Preliminary data analysis

3.1 Sample size

train_data %>%
  count(name = "Sample size") %>% 
  mutate(Data = "Optum training set") %>%
  select(Data, `Sample size`) %>%
  kable() %>%
  kable_styling()

Data	Sample size
Optum training set	13202

3.2 Missing data

missing_df <- train_data %>%
  select(one_of(vars$var)) %>%
  mutate_all(function (x) ifelse(is.na(x), 1, 0)) %>%
  mutate(id = factor(1:n())) %>%
  pivot_longer(cols = vars$var, names_to = "var", values_to = "missing") %>%
  left_join(vars, by = "var")

3.2.1 Proportion missing

# Compute proportion missing
prop_missing <- missing_df %>%
  group_by(varlab) %>%
  summarise(prop = mean(missing))

## `summarise()` ungrouping output (override with `.groups` argument)

# Plot
ggplot(prop_missing, aes(x = varlab, y = prop)) +
  geom_bar(stat = "identity") + 
  geom_text(aes(label = formatC(prop, format = "f", digits = 2)),
          nudge_y = .03, size = 3) +
ylim(c(0, 1)) +
xlab("") +
ylab("Proportion") +
coord_flip() +
scale_x_discrete(limits = rev(sort(vars$varlab)))

3.2.2 Missing by patient

ggplot(missing_df,
       aes(x = id, y = varlab,  fill = factor(missing))) +
  geom_raster() + 
  theme(axis.title.x = element_blank(),
        axis.text.x = element_blank(),
        axis.ticks.x = element_blank(),
        axis.title.y = element_blank(),
        legend.position = "bottom") +
  scale_fill_manual(name = "Missing",
                    values = c("lightgrey", "steelblue"),
                    labels = c("No", "Yes")) +
  scale_y_discrete(limits = rev(sort(vars$varlab)))

## Warning: Raster pixels are placed at uneven vertical intervals and will be shifted. Consider using geom_tile() instead.

missing_df %>% 
  # Count of missing by patient
  group_by(id) %>%
  summarise(n_missing = sum(missing),
            prop_missing = n_missing/n()) %>%
  
  # Plot
 ggplot(aes(x = prop_missing)) +
  geom_histogram(binwidth = .03, color = "white") +
  scale_x_continuous(breaks = seq(0, 1, .05)) +
  scale_y_continuous(n.breaks = 20) +
  xlab("Proportion of covariates that are missing") +
  ylab("Count") 

## `summarise()` ungrouping output (override with `.groups` argument)

3.3 Distributions

3.3.1 Covariates

3.3.1.1 Categorical

cat_var_df <-  train_data %>%
  select(one_of("ptid", cat_vars$var)) %>%
  pivot_longer(cols = cat_vars$var, names_to = "var", values_to = "value") %>%
  left_join(cat_vars, by = "var") %>%
  filter(!is.na(value)) %>%
  group_by(var, varlab, value) %>%
  summarise(n = n()) %>%
  group_by(varlab) %>%
  mutate(freq = n / sum(n)) %>%
  ungroup() %>%
  mutate(
    nudge_x = case_when(
      freq < 0.5 ~ 0.15,
      TRUE ~ -0.15
    )
  )

## `summarise()` regrouping output by 'var', 'varlab' (override with `.groups` argument)

ggplot(cat_var_df,
       aes(x = freq, y = value)) +
  geom_point() + 
  geom_text(aes(label = formatC(freq, format = "f", digits = 2)),
            nudge_x = cat_var_df$nudge_x, size = 3.75) +
  facet_wrap(~varlab, scales = "free_y", ncol = 4) +
  xlim(0, 1) +
  xlab("Proportion") +
  ylab("") +
  theme(axis.text.x = element_text(size = 10),
        axis.text.y = element_text(size = 10),
        strip.text.x = element_text(size = 9))

3.3.1.2 Continuous

3.3.1.2.1 Box plot

pivot_continuous_longer <- function(data, vars){
  col_names <- vars$var
  train_data %>%
    select(one_of("ptid", col_names)) %>%
    pivot_longer(cols = col_names, 
                 names_to = "var", 
                 values_to = "value") %>%
    left_join(vars, by = "var") %>%
    filter(!is.na(value)) 
}
continuous_var_df <- pivot_continuous_longer(train_data, 
                                             vars = continuous_vars)

## Note: Using an external vector in selections is ambiguous.
## ℹ Use `all_of(col_names)` instead of `col_names` to silence this message.
## ℹ See <https://tidyselect.r-lib.org/reference/faq-external-vector.html>.
## This message is displayed once per session.

plot_box <- function(data){
  ggplot(data, 
       aes(x = varlab, y = value)) +
  geom_boxplot(outlier.size = 1) + 
  facet_wrap(~varlab, scales = "free") +
  xlab("") +
  ylab("Value") +
   theme(axis.title.x=element_blank(),
        axis.text.x=element_blank(),
        axis.ticks.x=element_blank(),
        strip.text = element_text(size = 7))
}
plot_box(continuous_var_df)

3.3.1.2.2 Histogram

plot_hist <- function(data){
  ggplot(data,
       aes(x = value)) +
  geom_histogram(bins = 40, color = "white") +
  facet_wrap(~varlab, scales = "free") + 
  xlab("") + ylab("Frequency")
}
plot_hist(continuous_var_df)

3.3.1.2.3 Outliers

outer_fence <- function(v){
  q1 <- quantile(v, .25, na.rm = TRUE)
  q3 <- quantile(v, .75, na.rm = TRUE)
  iq <- (q3 - q1)
  return(as.numeric(q3 + 3 * iq))
}

format_percent <- function(x){
  paste0(formatC(100 * x, format = "f", digits = 1), "%")
}

train_data %>%
  select(one_of(lab_vars$var)) %>%
  pivot_longer(cols = lab_vars$var, names_to = "Lab") %>%
  filter(!is.na(value)) %>%
  group_by(Lab) %>%
  summarise(Maximum = max(value),
            `99%` = quantile(value, .99), 
            `Outer fence` = outer_fence(value),
            `% above outer fence` = format_percent(mean(value > outer_fence(value)))) %>%
  mutate(Lab = get_var_labs(Lab)) %>%
  kable() %>%
  kable_styling()

## `summarise()` ungrouping output (override with `.groups` argument)

Lab	Maximum	99%	Outer fence	% above outer fence
Alanine aminotransferase (ALT)	3017.00	229.63000	130.000	3.4%
Aspartate aminotransferase (AST)	7000.01	338.73000	157.000	3.6%
Creatinine	27.35	11.04000	3.280	6.5%
C-reactive protein (CRP)	638.00	359.09100	458.000	0.1%
Fibrin D-Dimer	114475.00	19510.80000	5030.000	6.8%
Ferritin	100000.01	7753.21000	3661.975	3.4%
Lactate dehydrogenase (LDH)	14007.00	1257.75000	1051.000	1.9%
Lymphocyte count	120.35	3.67930	3.530	1.1%
Neutrophil count	82.40	18.90000	18.290	1.2%
Procalcitonin	753.66	30.31320	1.270	10.2%
Platelet count (PLT)	1213.00	531.89000	569.000	0.7%
Oxygen saturation	100.00	100.00000	107.000	0.0%
Troponin I	72.47	2.30530	0.174	8.1%
White blood cell count (WBC)	127.50	22.53165	21.630	1.2%

# Truncate labs using outer fence
truncate_max <- function(v) outer_fence(v)

for (i in 1:length(lab_vars$var)){
  original_var <- lab_vars$var[i]
  truncated_var <- paste0(original_var, "_t")
  truncated_max_i <- truncate_max(train_data[[original_var]])
  train_data <- train_data %>% mutate(
    !!truncated_var := ifelse(get(original_var) > truncated_max_i, 
                              truncated_max_i, 
                              get(original_var))
    )
}

lab_vars_t <- lab_vars %>%
  mutate(var = paste0(var, "_t"))
vars <- bind_rows(vars, lab_vars_t) 

continuous_vars_t <- bind_rows(
  demographic_continuous_vars,
  vital_continuous_vars,
  lab_vars_t
)

continuous_var_t_df <- pivot_continuous_longer(train_data, 
                                               vars = continuous_vars_t)
plot_hist(continuous_var_t_df)

3.3.1.2.4 Scatterplots

x_vars <- c("age")
y_vars <- c("bmi")
p <- vector(mode = "list", length = length(x_vars))
for (i in 1:length(x_vars)){
  p[[i]] <- ggplot(train_data, aes_string(x = x_vars[i], y = y_vars[i])) + 
  geom_point() + 
  #geom_smooth(method = "loess", formula = y ~ x) +
  xlab(get_var_labs(x_vars[i])) +
  ylab(get_var_labs(y_vars[i]))
}
do.call("grid.arrange", c(p, nrow = 1))

## Warning: Removed 1551 rows containing missing values (geom_point).

3.3.1.3 Correlations with race

3.3.1.3.1 Race and age

ggplot(train_data, 
       aes(x = race, y = age)) +
  geom_boxplot(outlier.size = 1) + 
  xlab("") +
  ylab(get_var_labs("age")) 

3.3.1.3.2 Race and comorbidities

train_data %>%
  group_by(race) %>%
  summarise(`Unweighted` = mean(score),
            `Weighted` = mean(wscore)) %>%
  pivot_longer(cols = c("Unweighted", "Weighted"),
               names_to = "score_type") %>%
  
  # Plot
  ggplot(aes(x = race, y = value)) +
    geom_bar(stat = "identity") +
    facet_wrap(~score_type, nrow = 2) +
    xlab("") +
    ylab("Mean Charlson score") +
    scale_fill_discrete("") 

## `summarise()` ungrouping output (override with `.groups` argument)

train_data %>%
  # Tidy table
  filter(!is.na(race)) %>%
  select(one_of(c("ptid", "race", comorbidity_vars$var))) %>%
  pivot_longer(cols = comorbidity_vars$var, names_to = "comorbidity") %>%
  left_join(comorbidity_vars %>% rename(comorbidity_lab = varlab),
            by = c("comorbidity" = "var")) %>%
  group_by(race, comorbidity_lab) %>%
  summarise(prop = mean(1 * (value == "Yes"))) %>%
  
  # Plot
  ggplot(aes(x = comorbidity_lab, y = prop, fill = race)) +
    geom_bar(stat = "identity", position = "dodge") +
    theme(axis.text.x = element_text(angle = 60, hjust = 1)) +
    xlab("") +
    ylab("Proportion") +
    scale_fill_discrete("")

## `summarise()` regrouping output by 'race' (override with `.groups` argument)

3.3.1.3.3 Race and region

train_data %>%
  # Tidy table
  group_by(race, division) %>%
  tally() %>%
  group_by(race) %>%
  mutate(prop = n/sum(n)) %>%
  
  # Plot
  ggplot(aes(x = division, y = prop)) +
    geom_bar(stat = "identity") +
    facet_wrap(~race) +
    theme(axis.text.x = element_text(angle = 60, hjust = 1)) +
    xlab("") +
    ylab("Proportion") 

3.3.2 Outcomes

train_data %>%
  count(died) %>%
  mutate(died = ifelse(died == 1, "Yes", "No"),
         prop = n/sum(n)) %>%
  ggplot(aes(x = died, y = prop)) +
  geom_bar(stat = "identity") +
  geom_text(aes(label = formatC(prop, format = "f", digits = 2)),
            nudge_y = .03, size = 3) +
  xlab("Died") +
  ylab("Proportion") 

3.3.2.1 Time to death

train_data %>%
  mutate(months_to_death = death_month - index_month) %>%
  filter(!is.na(months_to_death)) %>%
  group_by( months_to_death) %>%
  tally() %>%
  
  # Plot
  ggplot(aes(x = factor(months_to_death), y = n)) +
  geom_bar(stat = "identity", position = "dodge") +
  xlab("Months from index date to death") +
  ylab("Count")

3.4 Table 1

train_data <- train_data %>%
  add_race_ethnicity()

3.5 Transformations of continuous variables

fit_univariate_logit <- function(var, data){
  make_f <- function(rhs){
    as.formula(paste("died", rhs, sep =" ~ "))
  }
  fit_logit <- function(f, data){
    glm(f, family = "binomial", data = data)
  }
  
  list(
    `Linear` = fit_logit(make_f(var), data),
    `Spline 3 knots` = fit_logit(make_f(sprintf("ns(%s, 5)", var)), data),
    `Spline 4 knots` = fit_logit(make_f(sprintf("ns(%s, 6)", var)), data),
    `Spline 5 knots` = fit_logit(make_f(sprintf("ns(%s, 7)", var)), data)
  )
}

predict_univariate_logit <- function(models, var, var_values, type = "response"){
  newdata <- data.frame(var = var_values)
  colnames(newdata) <- var
  map_dfc(models,
         function(x) predict(x, newdata = newdata, type = type)) %>%
  mutate(var = var_values) %>%
  pivot_longer(cols = names(models),
               names_to = "Model",
               values_to = "y")
}

midpoint <- function(x, digits = 2){
  lower <- as.numeric(gsub(",.*", "", gsub("\\(|\\[|\\)|\\]", "", x)))
  upper <- as.numeric(gsub(".*,", "", gsub("\\(|\\[|\\)|\\]", "", x)))
  return(round(lower+(upper-lower)/2, digits))
}

bin_y <- function(var, var_values){
  data <- train_data[, c(var, "died")] %>%
    filter(!is.na(get(var)))
  data <- data %>%
    mutate(x_cat = cut(get(var), breaks = 20),
           x_midpoint = midpoint(x_cat)) %>%
    group_by(x_midpoint) %>%
    summarise(y = mean(died),
              n = n())
  colnames(data)[1] <- var
  return(data)
}

plot_univariate_logit <- function(models, var, var_values, var_lab = "Variable",
                                  type = "response", ylab = "Probability of death"){
  # Plotting data
  predicted_probs <- predict_univariate_logit(models, var, var_values, type = type)
  ylab <- switch(type,
                 "link" = "Log odds",
                 "response" = "Probability of death",
                 stop("Type must be 'link' or 'response'")
  )
  binned_y <- bin_y(var, var_values)
  if (type == "link"){
    binned_y$y <- ifelse(binned_y$y == 0, .001, binned_y$y)
    binned_y$y <- ifelse(binned_y$y == 1, .99, binned_y$y)
    binned_y$y <- qlogis(binned_y$y)
  }
  
  # Plotting scales
  y_min <- min(c(binned_y$y, predicted_probs$y))
  y_max <- max(c(binned_y$y, predicted_probs$y))
  size_breaks <- seq(min(binned_y$n), max(binned_y$n),
                     length.out = 6)
  
  # Plot
  ggplot(predicted_probs,
       aes(x = var, y = y)) +
    geom_line() +
    geom_point(data = binned_y, aes_string(x = var, y = "y", size = "n")) +
    facet_wrap(~Model, ncol = 2) +
    xlab(var_lab) +
    ylab(ylab) +
    ylim(floor(y_min), ceiling(y_max)) +
    scale_size(name = "Sample size", range = c(0.3, 3), 
               breaks = round(size_breaks, 0)) 
}

make_seq <- function(var){
  var_min <- min(train_data[[var]], na.rm = TRUE)
  var_max <- max(train_data[[var]], na.rm = TRUE)
  seq(var_min, var_max, length.out = 100)
}

evaluate_univariate_logit <- function(var, print = TRUE){
  var_values = make_seq(var)
  var_lab = get_var_labs(var)
  
  # Do evaluations
  fits <- fit_univariate_logit(var, data = train_data)
  p_link <- plot_univariate_logit(fits, var, var_values, var_lab, type = "link")
  p_probs <- plot_univariate_logit(fits, var, var_values, var_lab, type = "response")
  bic <- sapply(fits, BIC)
  
  # Print and return
  if (print){
    print(p_link)
    print(p_probs)
    print(bic)
  }
  return(list(fits = fits, p_link = p_link, p_probs = p_probs,
              bic = bic))
}

3.5.1 Demographics

3.5.1.1 Age

uv_age <- evaluate_univariate_logit("age")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       9824.579       9851.028       9860.544       9869.824

3.5.1.2 Calendar time

uv_calendar_time <- evaluate_univariate_logit("calendar_time")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       11537.38       11557.18       11566.44       11574.07

3.5.2 Vitals

3.5.2.1 Body Mass Index

uv_bmi <- evaluate_univariate_logit("bmi")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10292.29       10288.69       10297.49       10302.76

3.5.2.2 Diastolic blood pressure

uv_dbp <- evaluate_univariate_logit("dbp")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10634.15       10640.63       10650.69       10656.71

3.5.2.3 Systolic blood pressure

uv_sbp <- evaluate_univariate_logit("sbp")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       11029.39       10836.81       10845.17       10854.40

3.5.2.4 Heart rate

uv_hr <- evaluate_univariate_logit("hr")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       11124.33       11040.09       11049.47       11058.91

3.5.2.5 Respiration rate

uv_resp <- evaluate_univariate_logit("resp")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10312.98       10067.79       10077.26       10087.38

3.5.2.6 Temperature

uv_temp <- evaluate_univariate_logit("temp")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       11230.95       10963.68       10971.20       10980.51

3.5.3 Labs

3.5.3.1 Alanine aminotransferase (U/L)

uv_alt <- evaluate_univariate_logit("alt")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       9857.797       9886.913       9893.624       9898.861

uv_alt_t <- evaluate_univariate_logit("alt_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       9855.126       9889.227       9898.667       9906.320

3.5.3.2 Alanine aminotransferase (U/L)

uv_ast <- evaluate_univariate_logit("ast")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       9729.408       9578.856       9586.813       9594.616

uv_ast_t <- evaluate_univariate_logit("ast_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       9589.601       9583.281       9592.101       9601.301

3.5.3.3 C-reactive protein (mg/L)

uv_crp <- evaluate_univariate_logit("crp")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       7538.319       7530.642       7539.116       7545.301

uv_crp_t <- evaluate_univariate_logit("crp_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       7536.665       7530.517       7538.984       7545.047

3.5.3.4 Creatinine (mg/dL)

uv_creatinine <- evaluate_univariate_logit("creatinine")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10654.48       10193.85       10187.69       10189.80

uv_creatinine_t <- evaluate_univariate_logit("creatinine_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10282.04       10163.87       10173.34       10180.45

3.5.3.5 Ferritin (ng/mL)

uv_ferritin <- evaluate_univariate_logit("ferritin")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       7274.299       7173.300       7184.291       7189.920

uv_ferritin_t <- evaluate_univariate_logit("ferritin_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       7186.515       7175.916       7187.092       7191.102

3.5.3.6 Fibrin D-Dimer (ng/mL)

uv_d_dimer <- evaluate_univariate_logit("d_dimer")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       1199.693       1187.102       1191.181       1195.529

uv_d_dimer_t <- evaluate_univariate_logit("d_dimer_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       1152.919       1176.244       1183.356       1190.450

3.5.3.7 Lactate dehydrogenase (U/L)

uv_ldh <- evaluate_univariate_logit("ldh")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       6861.909       6679.183       6682.554       6688.124

uv_ldh_t <- evaluate_univariate_logit("ldh_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       6677.667       6664.075       6672.957       6681.556

3.5.3.8 Lymphocyte count (10^3/uL)

uv_lymphocyte <- evaluate_univariate_logit("lymphocyte")

## Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10678.36       10360.50       10368.48       10377.26

uv_lymphocyte_t <- evaluate_univariate_logit("lymphocyte_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10503.39       10354.16       10362.34       10371.29

3.5.3.9 Neutrophil count (10^3/uL)

uv_neutrophil <- evaluate_univariate_logit("neutrophil")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10397.19       10378.45       10380.84       10388.58

uv_neutrophil_t <- evaluate_univariate_logit("neutrophil_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10363.36       10366.13       10372.90       10380.99

3.5.3.10 Oxygen saturation (%)

uv_spo2 <- evaluate_univariate_logit("spo2")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10563.49       10568.58       10577.87       10585.05

3.5.3.11 Procalcitonin (ng/mL)

uv_pct <- evaluate_univariate_logit("pct")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       6815.976       6379.961       6380.174       6385.194

uv_pct_t <- evaluate_univariate_logit("pct_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       6434.158       6372.393       6374.694       6383.237

3.5.3.12 Troponin I (ng/mL)

uv_tni <- evaluate_univariate_logit("tni")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       7532.792       6846.952       6854.298       6862.907

uv_tni_t <- evaluate_univariate_logit("tni_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       6876.227       6751.771       6761.990       6770.322

3.5.3.13 Platelet count (10^3/uL)

uv_plt <- evaluate_univariate_logit("plt")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10762.33       10739.65       10747.87       10756.42

uv_plt_t <- evaluate_univariate_logit("plt_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10758.79       10739.44       10747.63       10756.18

3.5.3.14 White blood cell count (10^3/uL)

uv_wbc <- evaluate_univariate_logit("wbc")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10634.83       10589.11       10592.99       10599.51

uv_wbc_t <- evaluate_univariate_logit("wbc_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10579.27       10572.41       10580.40       10588.15

4 Data preprocessing

4.1 Candidate variables for modeling

Let’s specify variables that will be candidates for our model and used during variable selection. We will combine race and ethnicity into a single variable, but will wait to do this until after multiple imputation (see next section).

vitals_to_include <- c("bmi", "smoke", "temp", "hr", "resp", "sbp")
labs_to_include <- c("spo2", "crp_t", "tni_t", "alt_t", "ast_t", "ferritin_t",
                     "creatinine_t", "ldh_t", "lymphocyte_t", "neutrophil_t",
                     "plt_t", "wbc_t")
vars <- vars %>%
  mutate(include = ifelse(group %in% c("Demographics", "Comorbidities"),
                          1, 0),
         include = ifelse(var %in% c(vitals_to_include, labs_to_include),
                          1, include))
get_included_vars <- function(){
  vars[vars$include == 1, ]$var
}

make_rhs <- function(vars){
  as.formula(paste0("~", paste(vars, collapse = " + ")))
}
candidate_model_rhs <- make_rhs(get_included_vars()) 

4.2 Missing data imputation

Note that there is no “Other” geographic region as all 9 of the US geographic divisions are included in the data. We will consequently recode this category to missing.

train_data <- train_data %>%
  mutate(division = ifelse(division == "Other", NA, division),
         division = ifelse(division %in% c("East South Central", "Mountain"),
                           "Other", division))

Let’s re-level them so that we have preferred reference categories.

train_data <- train_data %>% mutate(
  sex = relevel(factor(sex), ref = "Male"),
  division = relevel(factor(division), ref = "Pacific"),
  smoke = relevel(factor(smoke), ref = "Never"),
  bmi_cat = relevel(factor(bmi_cat), ref = "Normal")
)

We can now multiply impute data use MICE.

# Run MICE algorithm
mice_out <- train_data %>%
  select(c(one_of(get_included_vars()), "died")) %>%
  mutate_if(is.character, as.factor) %>%
  mice(m = 2, maxit = 5) 

## 
##  iter imp variable
##   1   1  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  alt_t  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   1   2  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  alt_t  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   2   1  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  alt_t  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   2   2  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  alt_t  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   3   1  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  alt_t  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   3   2  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  alt_t  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   4   1  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  alt_t  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   4   2  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  alt_t  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   5   1  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  alt_t  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   5   2  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  alt_t  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t

# Append datasets and add outcome
mi_df <- complete(mice_out, action = "long", include = TRUE) %>%
  as_tibble() %>%
  mutate(died = rep(train_data$died, mice_out$m + 1))

# To compare MICE to aregImpute
# areg_out <-  aregImpute(update.formula(candidate_model_rhs, ~.), 
#                                        n.impute = 2, data = train_data)

4.3 Imputation diagnostics

We will compare the distributions of the imputed and observed data.

make_imp_df <- function(object){
  # Get imputations
  if (inherits(object, "mids")){
    imp <- object$imp
  } else{ # aregImpute
   imp <- object$imputed 
   for (i in 1:length(imp)){
     cat_levels_i <- object$cat.levels[[i]]
     if (!is.null(cat_levels_i) && !is.null(imp[[i]])){
       levels <- sort(unique(c(imp[[i]])))
       imp[[i]] <- apply(imp[[i]], 
                         2, 
                         function(x) factor(x, levels = levels,
                                            labels = cat_levels_i))
     }
   }
  }
  
  # Create list of data frames
  is_numeric <- sapply(imp, function (x) is.numeric(x[, 1]))
  continuous_df <- vector(mode = "list", length = sum(is_numeric))
  cat_df <- vector(mode = "list", length = sum(!is_numeric))
  continuous_cntr <- 1
  cat_cntr <- 1
  for (i in 1:length(imp)){
    if(!is.null(nrow(imp[[i]])) && nrow(imp[[i]]) > 0 ){
        imp_i_df <- data.frame(var = names(imp)[i],
                               imp = rep(1:ncol(imp[[i]]), each = nrow(imp[[i]])),
                               value = c(as.matrix(imp[[i]]))) %>% 
          as_tibble()
        
    } else{
      imp_i_df <- NULL
    }
    if (is_numeric[i]){
      continuous_df[[continuous_cntr]] <- imp_i_df
      continuous_cntr <- continuous_cntr + 1
    } else{
      cat_df[[cat_cntr]] <- imp_i_df
      cat_cntr <- cat_cntr + 1
    }
  } 
  
  # Row bind data frames
  continuous_df = bind_rows(continuous_df) %>%
    mutate(obs = "Imputed",
           varlab = get_var_labs(var)) 
  
  cat_df = bind_rows(cat_df) %>%
    mutate(obs = "Imputed",
           varlab = get_var_labs(var)) 
              
  # Return
  return(list(continuous = continuous_df,
              cat = cat_df))
}
imp_df <- make_imp_df(mice_out)
#imp_df <- make_imp_df(areg_out)

# Plot continuous variables
## Data for plotting
obsimp_df_continuous <- bind_rows(
  imp_df$continuous,
  continuous_var_df %>%
    select(var, value, varlab) %>%
    mutate(imp = 0, obs = "Observed")
) %>%
  mutate(imp = ifelse(imp == 0, "Observed", paste0("Imputation ", imp))) %>%
  filter(var %in% unique(imp_df$continuous$var))
  
## Plot
ggplot(obsimp_df_continuous,
       aes(x = value, col = imp)) +
  geom_density(position = "jitter") +
  facet_wrap(~varlab, scales = "free", ncol = 3) +
  xlab("") + ylab("Density") +
  scale_color_discrete(name = "") +
  theme(legend.position = "bottom")

# Plot categorical variables
## Data for plotting
obsimp_df_cat <- 
  bind_rows(
    imp_df$cat %>%
      group_by(var, varlab, value, imp) %>%
      summarise(n = n()) %>%
      group_by(var, varlab, imp) %>%
      mutate(freq = n / sum(n)),
    cat_var_df %>%
      select(var, value, varlab, n, freq) %>%
      mutate(imp = 0, obs = "Observed")
  ) %>%
    mutate(imp = ifelse(imp == 0, "Observed", paste0("Imputation ", imp))) %>%
    filter(var %in% unique(imp_df$cat$var))

## `summarise()` regrouping output by 'var', 'varlab', 'value' (override with `.groups` argument)

# Plot
ggplot(obsimp_df_cat, 
       aes(x = value, y = freq, fill = imp)) +
  geom_bar(position = "dodge", stat = "identity") +
  facet_wrap(~varlab, scales = "free_x") +
  scale_fill_discrete(name = "") +
  xlab("") +
  ylab("Proportion") +
  theme(legend.position = "bottom",
        axis.text.x = element_text(angle = 45, vjust = 1, hjust = 1))

4.4 Recoding categorical variables

We create the race/ethnicity variable after imputation of missing values of race and ethnicity. We will also combine the diabetes and hypertension variables

mi_df <- mi_df %>% 
  add_race_ethnicity() %>%
  mutate(
    diab = case_when(
      diabunc == "Yes" | diabwc == "Yes" ~ "Yes",
      TRUE ~ "No"
    ),
    hyp = case_when(
      hypc == "Yes" | hypunc == "Yes" ~ "Yes",
      TRUE ~ "No"
    )
)
prop.table(table(mi_df %>% filter(.imp == 0) %>% pull(diab)))

## 
##        No       Yes 
## 0.6608847 0.3391153

prop.table(table(mi_df %>% filter(.imp == 0) %>% pull(hyp)))

## 
##        No       Yes 
## 0.4129677 0.5870323

Now lets adjust the variables for our models using the combined race and ethnicity category.

vars <- bind_rows(
  vars,
  tibble(var = c("race_ethnicity", "diab", "hyp"),
         varlab = c("Race/ethnicity", "Diabetes", "Hypertension"),
         group = c("Demographics", "Comorbidities", "Comorbidities"),
         include = 1)
) %>%
  mutate(include = ifelse(var %in% c("race", "ethnicity", "diabunc", "diabwc", 
                                     "hypc", "hypunc"),
                          0, 
                          include))
  
candidate_model_rhs <- make_rhs(get_included_vars())

Finally, we will (i) add the new race/ethnicity variable to the “MICE” object and (ii) create a list of imputed datasets for analysis.

mice_out <- as.mids(mi_df)
mi_list <- mi_df %>% 
  filter(.imp > 0) %>% 
  split(list(.$.imp))

5 Variable selection

candidate_model_rhs <- candidate_model_rhs %>%
  update.formula( ~. + rcs(age, 3) - age +
                       rcs(calendar_time, 3) - calendar_time +
                       rcs(bmi, 3) - bmi +
                       rcs(temp, 3) - temp +
                       rcs(hr, 3) - hr +
                       rcs(resp, 3) - resp +
                       rcs(sbp, 3) - sbp +
                       rcs(spo2, 3) - spo2 +
                       rcs(crp_t, 3) - crp_t +
                       rcs(tni_t, 3) - tni_t +
                       rcs(alt_t, 3) - alt_t +
                       rcs(ast_t, 3) - ast_t +
                       rcs(creatinine_t, 3) - creatinine_t +
                       rcs(ferritin_t, 3) - ferritin_t +
                       rcs(ldh_t, 3) - ldh_t,
                       rcs(lymphocyte_t, 3) - lymphocyte_t +
                       rcs(neutrophil_t, 3) - neutrophil_t +
                       rcs(plt_t, 3) - plt_t +
                       rcs(wbc_t, 3) - wbc_t)

rename_rcs <- function(v){
  rcs_ind <- grep("rcs", v)
  v[rcs_ind] <- sub("rcs.*)", "", v[rcs_ind])
  return(v)
}

rename_terms <- function(v){
  v <- gsub(" ", "_", v)
  v <- gsub("-", "", v)
  v <- gsub("/", "", v)
  v <- rename_rcs(v)
  return(v)
}

make_x <- function(data){
  x <- model.matrix(candidate_model_rhs, data)
  assign <- attr(x, "assign")
  colnames(x) <- rename_terms(colnames(x))
  x <- x[, -1]
  attr(x, "assign") <- assign[-1]
  return(x)
}

# List of x and y for each imputed dataset
x <- mi_list %>% map(make_x)
y <- mi_list %>% map(function(x) x[["died"]])

terms <- read.csv("risk-factors-terms.csv") %>%
  left_join(vars[, c("var", "group")], by = "var")

get_term_labs <- function(v, term_name = "term"){
  terms$termlab[match(v, terms[[term_name]])]
}

match_terms_to_vars <- function(t){
  terms$var[match(t, terms$term)]
}

# Number of iterations
n_rep <- 2
n_folds <- 5
n_imputations <- length(mi_list)

# Threshold for variable inclusion
inclusion_threshold <- 0.9

# Matrix to store inclusion results
inclusion_sim <- matrix(0,  ncol = ncol(x[[1]]) + 1, 
                        nrow = n_rep * n_imputations)

# Groups
groups <- attr(x[[1]], "assign") 

# Convenience function to extract coefficients from Group-Lasso
coef_cv_oem <- function(object){
  coef <- object$oem.fit$beta$grp.lasso
  lse_ind <- which(object$lambda[[1]] == object$lambda.1se.models)
  return(coef[, lse_ind])
}

# Variable selection via Group-Lasso
cntr <- 1
for (i in 1:n_imputations){
  for (j in 1:n_rep){
    
    # Cross-validation
    cv_fit <-  cv.oem(x = x[[i]], y = y[[i]], 
                      penalty = "grp.lasso", 
                      groups = groups, 
                      family = "binomial",
                      type.measure = "deviance"
                      )
  
    # Count nonzero coefficients 
    inclusion_sim[cntr, which(coef_cv_oem(cv_fit) != 0)] <- 1
    
    # Iterate
    cntr <- cntr + 1
  } # End repeated CV loop
} # End imputation loop 

Now, let’s check the inclusion probabilities from the above repeated cross-validation steps

# Percentage of simulations each term is included
inclusion_sim <- inclusion_sim[, -1] # Remove intercept
colnames(inclusion_sim) = colnames(x[[1]])

inclusion_summary <- tibble(term = colnames(inclusion_sim), 
                            prob = apply(inclusion_sim, 2, mean)) 
model_terms <- inclusion_summary %>%
  filter(prob >= inclusion_threshold) %>%
  pull(term)

# Percentage of simulations each variable is included
inclusion_summary <- inclusion_summary %>%
  mutate(var = match_terms_to_vars(term)) %>%
  mutate(varlab = get_var_labs(var)) %>% 
  distinct(prob, var, varlab)

# Plot
ggplot(inclusion_summary,
       aes(x = reorder(varlab, prob), y = prob)) +
  geom_bar(stat = "identity") + 
  geom_hline(yintercept = inclusion_threshold, linetype = "dotted", 
                color = "red", size = 1) +
  ylab("Probability of inclusion") +
  coord_flip() +
  theme(axis.title.y = element_blank())

Correlation heatmap among the selected features

abs(cor(x[[1]][, model_terms])) %>%
  set_colnames(get_term_labs(colnames(.))) %>%
  set_rownames(get_term_labs(rownames(.))) %>%
  as.data.frame.table(responseName = "Correlation") %>%
  ggplot(aes(x = Var1, y = Var2, fill = Correlation)) + 
  geom_tile() + 
  theme(axis.text.x = element_text(angle = 90, vjust = 0.5, hjust = 1),
        axis.title = element_blank(),
        legend.position = "bottom")

6 Model interpretation

6.1 Model terms and data

We include variables in the model based on the group Lasso simulation implemented above.

vars_to_exclude <- inclusion_summary %>%
  filter(prob < inclusion_threshold) %>%
  pull(var)

remove_terms_from_rhs <- function(f, vars_to_exclude){
  # First convert formula to string separated by +
  f_string <- Reduce(paste, deparse(f))
  f_string <- gsub("~", "", f_string)
  f_string <- gsub(" ", "", f_string)
  
  # Then convert string to vector
  f_vec <-   unlist(strsplit(f_string, "\\+"))
  pattern_to_exclude <- paste(vars_to_exclude, collapse = "|")
  f_vec <- f_vec[!grepl(pattern_to_exclude, f_vec)]
  
  # Convert string back to formula
  f_new <- paste0("~", paste(f_vec, collapse = " + "))
  return(as.formula(f_new))
}

model_rhs <- remove_terms_from_rhs(candidate_model_rhs, vars_to_exclude)

label(mi_df) <- map(colnames(mi_df), 
                    function(x) label(mi_df[, x]) <- get_var_labs(x))
dd <- datadist(mi_df, adjto.cat = "first")
options(datadist = "dd")

6.2 Fit

f_logit_all <- update.formula(model_rhs, died ~ .)

# We will fit 4 models:
lrm_names <- c("Age only",
                "All demographics",
                "Demographics and comorbidities",
                "All variables")

## (1): fit_age: Only includes age
f_logit_age <- died ~ age 
lrm_fit_age <- fit.mult.impute(f_logit_age, fitter = lrm, xtrans = mice_out, pr = FALSE,
                                x = TRUE, y = TRUE)

## (2): fit_d: All demographics including age
f_logit_d <- update.formula(f_logit_age, ~. + sex + rcs(calendar_time, 3) + 
                              race_ethnicity + division)
lrm_fit_d <- fit.mult.impute(f_logit_d, fitter = lrm, xtrans = mice_out, pr = FALSE,
                             x = TRUE, y = TRUE)

## (3): fit_dc: Demographics and comorbidities
c_vars <- vars %>%
  filter(group == "Comorbidities" & include == 1 & !var %in% vars_to_exclude) %>%
  pull(var)
f_logit_dc <- update.formula(f_logit_d, 
                             as.formula(paste0("~.+", paste(c_vars, collapse = "+"))))
lrm_fit_dc <- fit.mult.impute(f_logit_dc, fitter = lrm, xtrans = mice_out, pr = FALSE,
                              x = TRUE, y = TRUE)

## (4): fit: The main model including demographics, comorbidities, vitals, and labs
lrm_fit_all <- fit.mult.impute(f_logit_all, fitter = lrm, xtrans = mice_out, 
                               pr = FALSE, x = TRUE, y = TRUE)

# Note that we can also estimate the models with stats::glm
glm_fits_all <- mi_list %>%
  map(function (x) glm(f_logit_all, data = x, family = "binomial")) 
glm_fit_all <- glm_fits_all %>% pool()

6.3 Collinearity

For categorical and continous variables, we use anova model; for categorical and categorical variables, we use Cramer’s V; for continous and continous variables, we use spearman correlation.

## from https://stackoverflow.com/questions/52554336/plot-the-equivalent-of-correlation-matrix-for-factors-categorical-data-and-mi
mixed_assoc <- function(df, cor_method = "spearman", adjust_cramersv_bias = TRUE){
  df_comb <- expand.grid(names(df), names(df),  stringsAsFactors = F) %>% 
    set_names("X1", "X2")
  is_nominal <- function(x) inherits(x, c("factor", "character"))
  # https://community.rstudio.com/t/why-is-purr-is-numeric-deprecated/3559
  # https://github.com/r-lib/rlang/issues/781
  is_numeric <- function(x) { is.integer(x) || is_double(x)}
  f <- function(x_name, y_name) {
    x <-  pull(df, x_name)
    y <-  pull(df, y_name)
    result <- if(is_nominal(x) && is_nominal(y)){
        # use bias corrected cramersV as described in https://rdrr.io/cran/rcompanion/man/cramerV.html
        cv <- cramerV(as.character(x), as.character(y), 
                      bias.correct = adjust_cramersv_bias)
        data.frame(x_name, y_name, assoc = cv, type = "cramersV")
    } else if(is_numeric(x) && is_numeric(y)){
        correlation <- cor(x, y, method = cor_method, use = "complete.obs")
        data.frame(x_name, y_name, assoc = correlation, type = "correlation")
    } else if(is_numeric(x) && is_nominal(y)){
        # from https://stats.stackexchange.com/questions/119835/correlation-between-a-nominal-iv-and-a-continuous-dv-variable/124618#124618
        r_squared <- summary(lm(x ~ y))$r.squared
        data.frame(x_name, y_name, assoc = sqrt(r_squared), type = "anova")
    } else if(is_nominal(x) && is_numeric(y)){
        r_squared <- summary(lm(y ~ x))$r.squared
        data.frame(x_name, y_name, assoc = sqrt(r_squared), type = "anova")
    } else {
        warning(paste("unmatched column type combination: ", class(x), class(y)))
    }
    # finally add complete obs number and ratio to table
    result %>% 
      mutate(
        complete_obs_pairs = sum(!is.na(x) & !is.na(y)),
        complete_obs_ratio = complete_obs_pairs/length(x)) %>%
      rename(x = x_name, y = y_name)
  }
  # apply function to each variable combination
  map2_df(df_comb$X1, df_comb$X2, f)
}

# Create correlation matrix of associations
corr_mat <- mi_df %>% 
  filter(.imp == 1) %>% 
  select(any_of(get_included_vars())) %>%
  mixed_assoc() %>%
  select(x, y, assoc) %>%
  pivot_wider(names_from = y, values_from = assoc) %>%
  column_to_rownames("x") %>%
  as.matrix

# Make tile plot
m <- abs(corr_mat)
heatmap_df <- tibble(row = rownames(m)[row(m)], 
                     col = colnames(m)[col(m)], corr = c(m)) %>%
  mutate(row = get_var_labs(row),
         col = get_var_labs(col))
heatmap_df %>%
  ggplot(aes(x = row, y = col, fill = corr)) + 
  geom_tile() + 
  scale_fill_continuous("Correlation") +
  theme(axis.text.x = element_text(angle = 90, vjust = 0.5, hjust = 1),
        axis.title = element_blank()) 

6.4 Variable importance

# Compute variable important with Wald chi-square
lrm_anova_all <- anova(lrm_fit_all)

# Plot the result
lrm_anova_all %>% 
  as_tibble() %>%
  mutate(var = gsub(" ", "", rownames(lrm_anova_all)),
         varlab = get_var_labs(var),
         value = as.double(`Chi-Square` - `d.f.`)) %>%
  filter(!var %in% c("TOTAL", "Nonlinear", "TOTALNONLINEAR")) %>%

  ggplot(aes(x = value, y = reorder(varlab, value))) +
  geom_point() +
  theme(axis.title.y = element_blank()) +
  xlab(expression(chi^2-df)) 

6.5 Summary of odds ratios

6.5.1 Full model

lrm_summary_all <- summary(lrm_fit_all)

# Odds ratios
make_tidy_or <- function(object, model_name = NULL){
  if (is.null(model_name)) model_name <- "Model"
  object %>%
    as.data.frame() %>%
    as_tibble() %>%
    mutate(term = rownames(object),
           High = round(High, 0),
           Low = round(Low, 0),
           termlab = get_term_labs(term, "term2"),
           termlab = ifelse(!is.na(`Diff.`), 
                            paste0(termlab, " - ", High, ":", Low),
                            termlab),
           or = exp(Effect),
           or_lower = as.double(exp(`Lower 0.95`)),
           or_upper = exp(`Upper 0.95`)) %>%
    filter(Type == 1) %>%
    select(term, termlab, or, or_lower, or_upper) %>%
    arrange(-or) %>%
    mutate(model = model_name)
}
lrm_or_all <- make_tidy_or(lrm_summary_all, "All variables") 

# Odds ratio plot
ggplot(lrm_or_all, 
       aes(x = or, y = reorder(termlab, or))) +
  geom_point() +
  geom_errorbarh(aes(xmax = or_upper, xmin = or_lower,
                     height = .2)) +
  geom_vline(xintercept = 1, linetype = "dashed", col = "grey") + 
  theme(axis.title.y = element_blank()) +
  xlab("Odds ratio")

6.5.2 Comparison of model with and without vitals + labs

lrm_summary_dc <- summary(lrm_fit_dc)
lrm_or_dc <- make_tidy_or(lrm_summary_dc, "Demographics + comorbidities")

# Odds ratio comparison plot
lrm_or_comp <- bind_rows(lrm_or_all, lrm_or_dc) %>%
  filter(term %in% 
           terms[terms$group %in% c("Demographics", "Comorbidities"), ]$term2) %>%
  mutate(termlab = factor(termlab),
         termlab = reorder(termlab, or, function (x) -mean(x)))

ggplot(lrm_or_comp, 
       aes(x = termlab, y = or, col = model)) +
  geom_point(position = position_dodge(width = 1)) +
  geom_errorbar(aes(ymax = or_upper, ymin = or_lower,
                     width = .2), position = position_dodge(width = 1)) +
  facet_wrap(~termlab, strip.position = "left", ncol = 1, scales = "free_y") + 
  geom_hline(yintercept = 1, linetype = "dashed") + 
  theme(axis.title.y = element_blank()) +
  scale_color_discrete(name = "Model") +
  theme(axis.text.y = element_blank(),
        axis.ticks.y = element_blank(),
        strip.text.y.left = element_text(hjust = 0, vjust = 1, 
                                         angle = 0, size = 8),
        legend.position = "bottom") +
  ylab("Odds ratio") +
  coord_flip()

6.6 Predicted log odds

lrm_log_odds <- Predict(lrm_fit_all, ref.zero = TRUE)

# Get plotting data
p_log_odds <- ggplot(lrm_log_odds, sepdiscrete = "list")

# Continuous plot
log_odds_limit <- max(ceiling(abs(p_log_odds$continuous$data$yhat)))
log_odds_breaks <- seq(-log_odds_limit, log_odds_limit, 1)
p_log_odds_continuous <- p_log_odds$continuous$data %>%
  as_tibble() %>%
  mutate(varlab = get_var_labs(.predictor.)) %>%
  ggplot(aes(x = .xx., y = yhat)) +
  facet_wrap(~varlab, scales = "free_x", ncol = 4) +
  geom_line() +
  geom_ribbon(aes(ymin = lower, ymax = upper), alpha = 0.3) +
  geom_hline(yintercept = 0, linetype = "dashed", color = "grey") +
  ylab("Log odds") +
  scale_y_continuous(breaks = log_odds_breaks,
                     limits = c(-log_odds_limit, log_odds_limit)) +
  theme(axis.title.x = element_blank(), 
        strip.text = element_text(size = 7))

# Discrete plot
log_odds_limit <- max(ceiling(abs(p_log_odds$discrete$data$yhat)))
log_odds_breaks <- seq(-log_odds_limit, log_odds_limit, 1)
p_log_odds_discrete <- p_log_odds$discrete$data %>%
  as_tibble() %>%
  mutate(varlab = get_var_labs(.predictor.)) %>%
  ggplot(aes(x = yhat, y = .xx.)) +
  facet_wrap(~varlab, scales = "free_y", ncol = 4) +
  geom_point(size = .9) +
  geom_errorbarh(aes(xmin = lower , xmax = upper, height = 0)) +
  geom_vline(xintercept = 0, linetype = "dashed", color = "grey") +
  xlab("Log odds") +
  scale_x_continuous(breaks = log_odds_breaks,
                     limits = c(-log_odds_limit, log_odds_limit)) +
  theme(axis.title.y = element_blank(),
        strip.text = element_text(size = 7))

# Combine plots
grid.arrange(p_log_odds_discrete, p_log_odds_continuous,
             heights = c(5, 5))

## Warning: Removed 5 rows containing missing values (geom_errorbarh).

6.7 Predicted probabilities

# Make newdata
## Start with a random sample of patients
n_samples <- 1000
newdata <- mi_list[[1]] %>% 
  filter(.imp > 0) %>%
  sample_n(size = n_samples)
## Then create one version of the data for each sex, age, and calendar time 
## combination of interest
ages_to_vary <- seq(min(train_data$age),  max(train_data$age), 1)
sex_to_vary <- unique(train_data$sex)
min_index_date <- min(train_data$index_date)
calendar_times_to_vary <- c(as.Date("2020-03-01"), 
                            as.Date("2020-04-01"), 
                            as.Date("2020-05-01")) - min_index_date
vars_to_vary <- expand.grid(age = ages_to_vary,
                            sex = sex_to_vary[!is.na(sex_to_vary)],
                            calendar_time = as.numeric(calendar_times_to_vary))
newdata <- newdata %>% 
  select(-age, -sex, -calendar_time) %>%
  crossing(vars_to_vary)

# Predict
lrm_lp <- predict(lrm_fit_all, newdata = data.frame(newdata), se.fit = TRUE)
newdata <- newdata %>%
  mutate(lp = lrm_lp$linear.predictors,
         lp_se = lrm_lp$se.fit,
         prob = plogis(lp),
         lower = plogis(lp - qnorm(.975) * lp_se),
         upper = plogis(lp + qnorm(.975) * lp_se))
# Mean predictions by variables to vary
lrm_probs <- newdata %>%
  group_by(age, sex, calendar_time) %>%
  summarise(prob = mean(prob),
            lower = mean(lower),
            upper = mean(upper)) %>%
  mutate(date = min_index_date + calendar_time)

## `summarise()` regrouping output by 'age', 'sex' (override with `.groups` argument)

# Plot
ggplot(lrm_probs, aes(x = age, y = prob, color = sex, fill = sex)) +
  geom_line() +
  facet_wrap(~date) +
  # geom_ribbon(aes(ymin = lower, ymax = upper), alpha = 0.1,
  #             linetype = 0) +
  geom_line(aes(x = age, y = lower, color = sex),  linetype = "dotted") +
  geom_line(aes(x = age, y = upper, color = sex),  linetype = "dotted") +
  xlab("Age") +
  ylab("Mortality probability") +
  scale_color_discrete("") +
  scale_fill_discrete("") +
  theme(legend.position = "bottom")

7 Model validation

7.1 Optimism-adjusted model performance

We will start by using bootstrapping to estimate model performance and check for whether our in-sample fits are too optimistic. Specifically, we will use the following algorithm implemented in the rms package:

1. Calculate performance from the model fit on the original training sample, denoted as \(\theta_{orig}\).
2. For \(b = 1,\ldots,B\):
   1. Bootstrap (with replacement) from the original training sample.
   2. Fit the model to the bootstrapped data and estimate model performance, denoted as \(\theta_{train}\).
   3. Use the bootstrap fit to measure model performance on the original training sample, denoted as \(\theta_{test}\).
3. Calculate “optimism” as \(O = B^{-1} \sum_{b = 1}^B \theta_{b, train} - \theta_{b, test}\).
4. Calculate optimism-adjusted performance as \(\theta = \theta_{orig} - O\).

A shrinkage factor can also be estimated within each bootstrap sample to gauge the extent of overfitting. This is done by fitting \(g(Y) = \gamma_0 + \gamma_1 X\hat{\beta}\) where \(X\) and \(Y\) are the covariates and outcome, respectively, in the test sample (i.e., in step 2c) and \(\hat{\beta}\) is estimating in the training sample (i.e., step 2b). If there is no overfitting, then \(\gamma_0 = 0\) and \(\gamma_1 = 1\); conversely, if there is overfitting, then \(\gamma_1 < 1\) and \(\gamma_0 \neq 1\) to compensate.

lrm_val_age <- validate(lrm_fit_age)
lrm_val_d <- validate(lrm_fit_d)
lrm_val_dc <- validate(lrm_fit_dc)
lrm_val_all <- validate(lrm_fit_all) 

bind_cindex <- function(object){
  n_rows <- nrow(object)
  c_index <- (object["Dxy", 1:3] + 1)/2
  c_index[4] <- c_index[2] - c_index[3]
  c_index[5] <- c_index[1] - c_index[4]
  c_index[6] <- object[1, 6]
  
  return(rbind(object, c_index))
}

make_validation_tbl <- function(object){
  object %>% 
    bind_cindex() %>%
    set_colnames(c("(1) Original", "(2) Bootstrap training", 
                   "(3) Bootstrap test", "Optimism: (2) - (3)", 
                   "Original (corrected): (1) - (4)", "N")) %>%
    kable() %>%
    kable_styling()
}

make_validation_tbl(lrm_val_age)

	Original	Bootstrap training	Bootstrap test	Optimism: (2) - (3)	Original (corrected): (1) - (4)	N
Dxy	0.5489496	0.5473912	0.5489496	-0.0015584	0.5505081	40
R2	0.2120804	0.2105744	0.2120804	-0.0015060	0.2135865	40
Intercept	0.0000000	0.0000000	0.0126035	-0.0126035	0.0126035	40
Slope	1.0000000	1.0000000	1.0050614	-0.0050614	1.0050614	40
Emax	0.0000000	0.0000000	0.0035725	0.0035725	0.0035725	40
D	0.1319113	0.1306356	0.1319113	-0.0012757	0.1331871	40
U	-0.0001515	-0.0001515	-0.0000112	-0.0001403	-0.0000112	40
Q	0.1320628	0.1307871	0.1319225	-0.0011354	0.1331983	40
B	0.1161667	0.1159303	0.1161838	-0.0002535	0.1164203	40
g	1.3975928	1.3928700	1.3975928	-0.0047228	1.4023156	40
gp	0.1459953	0.1449631	0.1459953	-0.0010322	0.1470275	40
c_index	0.7744748	0.7736956	0.7744748	-0.0007792	0.7752540	40

make_validation_tbl(lrm_val_all)

	Original	Bootstrap training	Bootstrap test	Optimism: (2) - (3)	Original (corrected): (1) - (4)	N
Dxy	0.7872930	0.7904178	0.7832418	0.0071760	0.7801170	40
R2	0.4683287	0.4729904	0.4626797	0.0103107	0.4580180	40
Intercept	0.0000000	0.0000000	-0.0280367	0.0280367	-0.0280367	40
Slope	1.0000000	1.0000000	0.9753365	0.0246635	0.9753365	40
Emax	0.0000000	0.0000000	0.0104909	0.0104909	0.0104909	40
D	0.3188166	0.3227661	0.3142967	0.0084694	0.3103472	40
U	-0.0001515	-0.0001515	0.0001081	-0.0002595	0.0001081	40
Q	0.3189681	0.3229176	0.3141886	0.0087289	0.3102392	40
B	0.0856761	0.0851932	0.0864409	-0.0012478	0.0869238	40
g	2.4613585	2.4843901	2.4197283	0.0646618	2.3966967	40
gp	0.2078073	0.2088491	0.2065885	0.0022607	0.2055466	40
c_index	0.8936465	0.8952089	0.8916209	0.0035880	0.8900585	40

summarize_performance <- function(objects){
  metrics <- c("Dxy", "R2", "B")
  map_df(objects, function (x) x[metrics, "index.orig"],
                .id = "Model") %>%
    mutate(`C-index` = (Dxy + 1)/2) %>%
    rename(`Brier score` = B, `R-Squared` = R2) %>%
    select(-Dxy) %>%
    kable(digits = 3) %>%
    kable_styling()
}

list(lrm_val_age, lrm_val_d, lrm_val_dc, lrm_val_all) %>%
  set_names(lrm_names) %>%
  summarize_performance() 

Model	R-Squared	Brier score	C-index
Age only	0.212	0.116	0.774
All demographics	0.229	0.114	0.785
Demographics and comorbidities	0.257	0.112	0.803
All variables	0.468	0.086	0.894

7.2 Calibration curves

lrm_cal_age <- calibrate(lrm_fit_age)
lrm_cal_d <- calibrate(lrm_fit_d)
lrm_cal_dc <- calibrate(lrm_fit_dc)
lrm_cal_all <- calibrate(lrm_fit_all)
lrm_cal_list <- list(lrm_cal_age, lrm_cal_d, lrm_cal_dc, lrm_cal_all)
names(lrm_cal_list) <- lrm_names

plot_calibration <- function(object){
  # Make tibble
  cal_df <- map2(object, names(object), function(x, y){
    x[, ] %>%
      as_tibble() %>%
      mutate(model = y)
  }) %>% 
    bind_rows() %>%
    mutate(model = factor(model, levels = lrm_names))
  
  
  # Plot
  breaks <- seq(0, 1, .2)

  ggplot() + 
    geom_line(data = cal_df, mapping = aes(x = predy, y = calibrated.orig, 
                                           color = "Apparent")) + 
    geom_line(data = cal_df, mapping = aes(x = predy, y = calibrated.corrected,
                                           color = "Bias-corrected")) +
    geom_abline(intercept = 0, slope = 1, linetype = "dashed", color = "grey") +
    facet_wrap(~model) +
    scale_x_continuous(breaks = breaks, limits = c(0, 1)) +
    scale_y_continuous(breaks = breaks, limits = c(0, 1)) +
    xlab("Predicted probability") +
    ylab("Actual probability") +
    scale_colour_manual(name = "",
                        values = c("Apparent" = "black", 
                                   "Bias-corrected" = "red")) +
    theme(legend.position = "bottom") 
}

plot_calibration(lrm_cal_list)

7.3 Performance on test set